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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 27 "MAPLE WORKSHEET for SERIE
S " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 452 "\nThis worksheet is intend
ed to provide a graphical interpretation of sequences and their limits
. The Series command provides the following information:\n           i
) numerical values for the first N partial sums of the series;\n      \+
    ii) the sum of the series, if it converges;\n         iii) a plot \+
of the first N partial sums of the series.\n\nREMEMBER: Do NOT confuse
 the sequence of partial sums and the sequence formed by the terms of \+
the series.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "The simplest way to figure out sum
 of an infinite series is by using 'sum' Maple syntax below. " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "sum(1/n^2, n=1..infinity);" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT -1 172 "The 'Sequence' and 'Serie
s' below are defined by a procesure. The first step is to define the M
aple procedures for displaying series (and sequences) in a convenient \+
manner." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 441 "Sequence := proc( expr,
 arg::name=range )\n   local f, i, ii, S, var, lo, hi, m, M;\n   var :
= lhs(arg); lo := op(1,rhs(arg) ); hi := op(2, rhs(arg) );\n   f := un
apply( expr, var );\n   S := [seq( [i, evalf(f(i))], i=lo..hi )];\n   \+
m := min( 0, seq( op(2,s), s=S ) );\n   M := max( 0, seq( op(2,s), s=S
 ) );\n   print( S );\n   print( Limit( f(var), var=infinity ) = limit
( f(ii), ii=infinity) );\n   plot( S, var=lo..hi, y=m..M, style=POINT \+
);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 529 "Series := proc
( expr, arg::name=range )\n   local f, i, ii, j, P, s, var, lo, hi, m,
 M;\n   var := lhs(arg); lo := op(1,rhs(arg) ); hi := op(2, rhs(arg) )
;\n   f := unapply( expr, var );\n   s := 0; P := NULL;\n   for i from
 lo to hi do\n      s := s+f(i);\n      P := P, [i, evalf(s)];\n   od;
\n   P := [P];\n   m := min( 0, seq( op(2,p), p=P ) );\n   M := max( 0
, seq( op(2,p), p=P ) );\n   print( P );\n   print( Sum( f(var),var=lo
..infinity ) = evalf( sum( f(ii), ii=lo..infinity ) ) );\n   plot( P, \+
var=lo..hi, y=m..M, style=POINT );\nend:" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 224 "The first few examples focus on the distinction between \+
the sequence of partial sums and the sequence of the terms of the seri
es. First, here are the first few terms of the sequence formed by the \+
terms of the series \{a[k] \}." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "S
equence( 100*(1/2)^k, k=1..37 );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
147 "And, here are the first 50 partial sums (numerically and graphica
lly). Notice that, in this case, both sequences converge, but to diffe
rent limits." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Series( 100*(1/2)^k
, k=1..37 );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "Now, for some mo
re geometric series. First, here's a geometric series with r>1 (what s
hould happen here?):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Sequence( 2
^(k-1), k=1..10 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Serie
s( 2^(k-1), k=1.. 10 );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "This i
s an example of a divergent series." }}{PARA 0 "" 0 "" {TEXT -1 182 "T
his geometric series is also divergent, but in a different way. Note t
hat the sequence of terms alternates between 1 and -1; the sequence of
 partial sums alternates between 1 and 0." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "Sequence( (-1)^(k-1), k=1..10 );" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 30 "Series( (-1)^(k-1), k=1..10 );" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 144 "These two examples illustrate the n\{th
\} Term Test for the DIVERGENCE of a series. Since the terms do NOT co
nverge to 0, the series MUST DIVERGE." }}{PARA 0 "" 0 "" {TEXT -1 324 
"Think about what the n\{th\} Term Test is saying. A series converges \+
when you can add an infinite number of numbers to obtain a FINITE answ
er. If the individual terms do not become small (converge to 0), the s
equence of partial sums will keep changing by this (not small) amount \+
and so cannot be converging to a finite number." }}{PARA 0 "" 0 "" 
{TEXT -1 179 "The converse of the n\{th\} Term Test does NOT apply. Ju
st because the terms converge to 0 does NOT IMPLY the series converges
. The classical counter-example is the Harmonic Series:" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 25 "Sequence( 1/n, n=1..50 );" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 23 "Series( 1/n, n=1..40 );" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 309 "The Alternating Harmonic Series is a different st
ory. Here the terms converge to 0, and the series converges. Notice ho
w the partial sums bounce back and forth. With each partial sum, the a
mplitude of the oscillation decreases; ultimately, the sequence of par
tial sums does converge (but not to a nice value)." }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 28 "Series( (-1)^n/n, n=1..40 );" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 29 "Series( 2/(k-1)/k, k=3..85 );" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 912 "SUMMARY:\n\n
There have been three main points to this demonstration.\n\n1. Notice \+
the distinction between the sequence of partial sums and the sequence \+
of terms of the series. The partial sums are difficult to find, in gen
eral. We will be developing techniques which permit us to make a concl
usion about the sequence of partial sums (and\nhence the series) while
 examining only the terms of the sequence. Please, work hard to unders
tand this difference.\n\n2. The n\{th\} Term Test is a simple test tha
t enables us to examine the terms of the series to determine when some
 series diverge. This should be the first test that you apply to any s
eries. Here's another statement of the n\{th\} Term Test:\n   if lim a
[n] <> 0 then the series MUST DIVERGE\n   if lim a[n] = 0 then the ser
ies MAY CONVERGE (but some other test will have to be used to decide)
\n\n3. Geometric series converge only when the radius satisfies: -1 < \+
r < 1." }}}}{MARK "3 0 0" 87 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }